from The Century Dictionary and Cyclopedia

n. A literary composition, originally in verse, characterized by the expression of indignation, scorn, or contemptuous facetiousness, denouncing vice, folly, incapacity, or failure, and holding it up to reprobation or ridicule: a species of literary production cultivated by ancient Roman writers and in modern literature, and directed to the correction of corruption, abuses, or absurdities in religion, politics, law, society, and letters.

n. Hence, in general, the use, in either speaking or writing, of irony, sarcasm, ridicule, etc., in exposing, denouncing, or deriding vice, folly, indecorum, incapacity, or insincerity.

n. Vituperation; abuse; backbiting.

n. A satirist.

n.Synonyms Pasquinade, Invective, etc. See lampoon.

n. Irony, Sarcasm, Satire, ridicule. Irony may be of the nature of sarcasm, and sarcasm may possibly take the form of irony; but sarcasm is generally too severe, and therefore too direct, to take an ironical form; both may be means of satire. The essential thing about irony is the contradiction between the literal and the manifest meaning: as, “Is not a patron, my lord, one who looks with unconcern on a man struggling for life in the water, and, when he has reached the ground, encumbers him with help?” (Johnson, To Chesterfield.) “Irony … is the humorous wresting of language from its literal use for the expression of feeling, either happy or painful, but too vehement to be contented with that literal use. … When the thoughtful spirit of Macbeth is distorted by guilt, and as the agony of that guilt grows more and more intense, the pent-up misery either flows forth in a subdued irony or breaks out in that which is fierce and frenzied.” The essential thing about sarcasm is its cutting edge; it therefore is intensely concentrated, lying in a sentence or a phrase; it is used to scourge the follies or foibles or vices of men, but has little of reformatory purpose. Satire is more elaborate than sarcasm, is not necessarily bitter, and has, presumably, some aim at the reformation of that which is satirized. “Well-known instances of ironical argument are Burke's ‘Vindication of Natural Society,’ in which Bolingbroke's arguments against religious institutions are applied to civil society; Whately's ‘Historic Doubts,’ in which Hume's arguments against Christianity are used to prove the non-existence of Napoleon Bonaparte; Swift's ‘Argument against the Abolishment of Christianity,’ and his ‘Modest Proposal’ for relieving Ireland from famine by having the children cooked and eaten.”

Etymologies

(American Heritage® Dictionary of the English Language, Fourth Edition)

Implied in satiric (attested in 1387), from Latin satira, from earlier satura, from lanx satura ("full dish"), from feminine of satur. Altered in Latin by influence of Ancient Greek σάτυρος (saturos, "satyr"), on the mistaken notion that the form is related to the Greek σατυρικό δράμα (saturiko drama, "satyr drama"). (Wiktionary)

Examples

Of how much confusion the spelling which used to be so common, ‘satyr’ for ‘satire’, is at once the consequence, the expression, and again the cause; not indeed that this confusion first began with us {279}; for the same already found place in the Latin, where ‘satyricus’ was continually written for ‘satiricus’ out of a false assumption of the identity between the Roman _satire_ and the Greek _satyric_ drama.

The word "satire" coming from the Greek "satyr" plays which would be staged after a trilogy of tragedies, to finish the proceedings on a high, I think the modern satire is still following on in that tradition, an antidote to solemnity, a puncturing of the pomp.

More importantly, you ignore the fact that Greeks also had a third dramatic form, the satyr play (from which we derive the English word "satire"), which was very close to what we mean by modern comedy.

Even though most of the pointed satire is delivered through the story of the decidedly non-Messianic Brian (thereby avoiding blasphemy - kind of a technicality), the rampant fanaticism and hypocrisy that the Pythons poke fun at is universal.